Cubic Residue Symbols and Triple Milnor Invariants in Eisenstein Integers — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
FINDING: Dirichlet characters mod 3 connect to Eisenstein integers (ℚ(√−3)), whose prime angular distribution is tied to hexagonal lattice symmetry — the most relevant result is the arXiv paper on mod 3 triple Milnor invariants and cubic residue symbols in the Eisenstein field. | MATH: Eisenstein integers: ℤ[ω], ω = e^(2πi/3) = −1/2 + i√3/2. Norm: N(a+bω) = a² − ab + b². Cubic residue symbol: (α/π)₃ ∈ {1, ω, ω²}. Dirichlet characters mod 3: χ₀, χ₁ (trivial), χ₂ with χ₂(2) = −1, χ₂(1) = 1. Triple Milnor invariant: μ(1,2,3) mod 3. | CONNECTION: **Direct hexagonal symmetry.** The Eisenstein field has 6-fold rotational symmetry (roots of unity: ±1, ±ω, ±ω²). Prime angular distribution in ℚ(√−3) is anisotropic — primes split into 6 sectors (mod 6 classes) with equal density, but *individual* primes cluster along 6 rays in the complex plane (angles 0°, 60°, 120°, ...). The ratio 1/6 = 0.1667, and the *inverse* 6 = 2π/60° — a base-60 angular division. The cubic residue symbol takes values in
Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
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BibTeX
@misc{caldin2026cubic,
title = {{Cubic Residue Symbols and Triple Milnor Invariants in Eisenstein Integers — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23410015},
url = {https://zenodo.org/records/23410015},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cubic Residue Symbols and Triple Milnor Invariants in Eisenstein Integers — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23410015
UR - https://zenodo.org/records/23410015
AB - FINDING: Dirichlet characters mod 3 connect to Eisenstein integers (ℚ(√−3)), whose prime angular distribution is tied to hexagonal lattice symmetry — the most relevant result is the arXiv paper on mod 3 triple Milnor invariants and cubic residue symbols in the Eisenstein field. | MATH: Eisenstein integers: ℤ[ω], ω = e^(2πi/3) = −1/2 + i√3/2. Norm: N(a+bω) = a² − ab + b². Cubic residue symbol: (α/π)₃ ∈ {1, ω, ω²}. Dirichlet characters mod 3: χ₀, χ₁ (trivial), χ₂ with χ₂(2) = −1, χ₂(1) = 1. Triple Milnor invariant: μ(1,2,3) mod 3. | CONNECTION: **Direct hexagonal symmetry.** The Eisenstein field has 6-fold rotational symmetry (roots of unity: ±1, ±ω, ±ω²). Prime angular distribution in ℚ(√−3) is anisotropic — primes split into 6 sectors (mod 6 classes) with equal density, but *individual* primes cluster along 6 rays in the complex plane (angles 0°, 60°, 120°, ...). The ratio 1/6 = 0.1667, and the *inverse* 6 = 2π/60° — a base-60 angular division. The cubic residue symbol takes values in Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -